Stable models of Hecke operators

0.27 MB·English

by  Jan Vonk

STABLE MODELS OF HECKE OPERATORS
JANVONK
5
1 ABSTRACT. Weinvestigatethegeometryofcorrespondencesbetweencurves,andprovethatcorrespondences
0 overanon-Archimedeanvaluedfieldhavepotentiallystablereduction, generalisingandstrengtheningre-
2
sults of Coleman and Liu. This yields a concrete description of the operator on the cohomology of the

y generic fibres arising from linearisation of the correspondence, via the weight-monodromy filtration and
a Picard–Lefschetztheory. WeexplicitlydeterminestablemodelsofHeckeoperatorsonvariousquaternionic
M Shimuracurves, andproveageneralisationofthegeometrictheoryofcanonical subgroupsbyGorenand
Kassaei.

8
1

]
T
CONTENTS
N
. Introduction 1
h
t 1. Specialisationmapsandformalfibres 4
a
m 2. Stablemodelsofcorrespondences 6
[ 3. Weight-monodromyandskeletaofcorrespondences 8

2 4. HeckeoperatorsonShimuracurves 12
v
8 5. Overconvergent p-adicmodularforms 16
8
References 18
4
3
0
.
1
0 INTRODUCTION
5
1 Heckeoperators arise as linearisations of correspondences between modular curves. Classical results
:
v due to Eichler–Shimura, Mestre–Oesterlé and Ribet exhibit deep arithmetic information by studying
i reductionsmod pofvarious Heckecorrespondences. Thesemethodsareusuallyrestrictedtosituations
X
wherethelevelisdivisibleby patmostonce,astheycruciallyrelyonthesemi-stabilityofthecanonical
r
a Katz–Mazur models of the modular curves involved. Even though precise information on semi-stable
models of modular curves is known in great generality by the recent work of Weinstein [Wei12], we
need additional information about the degeneracy maps to carry out similar arguments. In this paper,
weusenon-Archimedean analytic geometrytoproposeasystematic framework fortheinvestigationof
operatorsarisingfromcorrespondencesbetweencurves,andgeneraliseseveralresultsintheliterature.
Wepostponelinearisationaslongaspossible,andstartbystudyingcorrespondencesasgeometricob-
jects. Weprovethatcorrespondencesbetweencurveshavepotentiallystablereduction,generalisingand
strengthening thework ofColeman [Col03] andLiu [Liu06],resulting in astrong analogue ofthethe-
orem of Deligne–Mumford [DM69]. We introduce skeletalsemi-stable models forcorrespondences, and
apply themtothestudyof theresulting spectrumvia theweight-monodromy filtration oncohomology.
1
2 JANVONK
This suggests a systematic geometric study of spectra of Hecke operators, avoiding the use of moduli
interpretations which might beabsent for automorphic forms that do not arise from PEL Shimura vari-
eties. Asanillustration ofourmethods,wegeneralisetheworkofGoren–Kassaei [GK06]oncanonical
subgroups,followingtheirstrategyoffocusingontheunderlyinggeometryoftheconsideredmorphisms.
Stable models of correspondences. Coleman [Col03] and Liu [Liu06] prove potentially semi-stable
reduction theorems for finite morphisms between curves over a general class of fields. We generalise
and strengthen their results to the setting of correspondences between curves, compactifying a suitably
definedmodulistack. Moreprecisely,letC:Y ←X →Y beacorrespondencebetweensmooth,proper,
1 2
geometricallyconnectedcurvesX,Y andY overanon-ArchimedeanfieldK,wherethemapsarefinite.
1 2
TheoremA. AfterafiniteseparablefieldextensionofK,wecanfindacorrespondenceC:
X
π π
1 2
Y Y
1 2
where X,Y and Y are semi-stable models over the valuation ring of K for X,Y and Y , the morphisms
1 2 1 2
π andπ arefinite,suchthatCrestrictstoConthegenericfibres. Thereisauniquesuchcorrespondence
1 2
whichisminimalwithrespecttotherelationofpairwisedominationofitsobjects.
Infact,weobtainaconsiderablystrongerstatementasacorollaryofTheoremB.Weshowthatwemay
always findaskeletalsuchcorrespondenceC, which satisfiestheadditional constraint thatthesingular
loci of thespecial fibres of Y and Y pull backto the singular locus of thespecial fibre of X. Skeleta
1 2
of correspondences may naturally be considered as tropical objects. The association of a skeleton to a
correspondencebetweencurvesmaybeviewedasananalogueofthetropicalisationmap
ad trop
Trop:M −→ M ,
g,n g,n
fromtheDeligne–Mumford–Knudsenmodulistackof n-pointedcurves ofgenus g tothecorresponding
moduli space of tropical curves. This map is studied by Abramovich, Caporaso and Payne [ACP15],
and is the retraction of the moduli stack of curves onto its canonical skeleton in the sense of Thuillier
[Thu07].
Weight-monodromy and Shimura curves. Skeletal semi-stable models give us precise information
about the spectral properties of various linearisations of C. For l a prime different from the residue
characteristicofK,weobtainalinearmap
C∗:Hi (Y ,Q )→Hi (Y ,Q ),
ét 1,K l ét 2,K l
defined by π ◦π∗. This map is strict in the sense that it respects the weight-monodromy filtrations,
2,∗ 1
andtheinducedmapongradedpiecesmaybedescribedexplicitlyfromaskeletalsemi-stablemodelvia
Picard–Lefschetztheory. NéroncomponentgroupsΦ aredescribedbythel-adicmonodromypairingon
i
cohomologygroups,andwemayconsequentlyalsocomputetheinducedmapC∗:Φ →Φ .
1 2
Applied to the setting of quaterionic Shimura curves, this allows us to revisit many classical results
suchastheEichler–Shimurarelation,theworkofRibetonlevellowering[Rib90],andgraphalgorithms
dueto Mestre–Oesterlé [Mes86]and Dembélé–Voight [DV13]. Theorems C andDexplicitly determine
skeletal semi-stable models at p of various Hecke operators T on quaternionic Shimura curves. This
l
is done both for l 6= p and l = p, and the latter case exhibits extra components not present in the
STABLEMODELSOFHECKEOPERATORS 3
stable models of the individual curves involved. Letting X denote a quaternionic Shimura curve with
sufficientlysmalllevelawayfromp,weobtaininsection4thefollowingpicturesforthestableskeleta:
T onX (p) T onX U onX (p)
l 0 p p 0
Overconvergent p-adic modular forms. Our initial goal was a geometric study of the Atkin opera-
tor U on spaces of overconvergent p-adic modular forms. Theorem E generalises a result of Coleman
p
[Col05],andstatesthatcertaincomponentsofU classifysupersingularcurvesthatfailtohaveacanon-
p
ical subgroup in the sense of Katz [Kat73] and Lubin [Lub79], or curves p-isogenous to them. These
components mark the boundary of the regions X studied in Coleman [Col97] for r = p/(p+1) and
r
r=1/(p+1). Consequently,weobtainanexplicitdescriptionofthe“boundarycorrespondence"induced
byU . WethengeneralisetheworkoncanonicalsubgroupsbyGoren–Kassaei[GK06],andfurthertheir
p
geometric line of investigation by proving the existence of a “canonical subgroup section” for general
morphisms:
Theorem F. Let g :X →Y be a finite flat map between semi-stable curves, and let g:X →Y be the
1 1
minimalskeletalsemi-stablemodeldominating g. If g isanisomorphism onacomponent Z ofX ,then g
s
isanisomorphismonsp−1(Z).
X1
Themorphismgiscalledtheskeletalstablehullof g. Theresulting sectionof g is maximal inaprecise
sense, and the true virtue of Theorem F is that it identifies this maximal section to g precisely, most
naturally phrased in terms of the skeletal stable hull of g. A particular case was handled by Goren–
Kassaei[GK06],wherethemaximalsectioncouldbeidentifiedbyadhocmethods.
Outline. Werecallresultsonthespecialisationmapofadicspacesandfunctorialityofsemi-stablemod-
elsofcurvesinsection1,andprovethatcorrespondencesofcurvesovernon-Archimedeanvaluedfields
have potentially semi-stable reduction in section 2. We explain in section 3 how the knowledge of a
skeletalsemi-stablemodelaidsusinthecomputationofthelinear mapbetweenétalecohomologyand
Néron component groups arising from linearisation of a correspondence. We explicitly determine the
stable models of Hecke operators on various quaternionic Shimura curves in section 4, and generalise
thegeometricargumentsofGoren–Kassaeiontheexistenceofcanonicalsubgroupsinsection5.
Acknowledgements. We are greatly indebted to Netan Dogra for countless enlightening discussions,
and to Minhyong Kim for proposing a study of the geometry of correspondences. We thank Alexan-
derBetts,K˛estutis Cˇesnaviˇcius, MatthewEmerton, Lorenzo Fantini andChristian Johanssonforseveral
helpful comments. The author is supported by the Engineering and Physical Sciences Research Coun-
cil (EPSRC), the Andrew Mullins Scholarship, St. Catherine’s College Oxford, and the Leatherseller’s
Company.
4 JANVONK
1. SPECIALISATION MAPS AND FORMAL FIBRES
We start by recalling the notion of semi-stable vertex sets of smooth quasi-projective curves. They
provide us with a combinatorial tool to understand semi-stable models and finite maps between them,
viathespecialisationmap. MostofthematerialinthissectionistreatedinmoredetailbyBaker–Payne–
Rabinoff [BPR11]andAmini–Baker–Brugallé–Rabinoff [ABBR15]using Berkovich geometry. Weadopt
thelanguageofadicspacesasdevelopedbyHuber[Hub93,Hub94],asitstreamlinesourpresentation.
1.1. Notation. LetKbeanalgebraicallyclosed,complete,non-Archimedeanfieldwithtopologyinduced
byanon-trivialvaluation|·|ofrank1. LetRbeitsvaluationringwithmaximalidealmandresiduefield
k. ForacurveX overR,wewriteX foritsspecialfibre, and X foritsgeneric fibre. Welet f :X →Y
s
be a finite morphism between smooth proper connected curves over K, and Xad,Yad the adic spaces
associated to X,Y. We will abuse notation and write f for the induced map fad between these adic
spaces. Givenapoint x ∈Xad(K),wewrite k(x)fortheresiduefieldofOXad,x and k(x)+ fortheimage
ofO+ .
Xad,x
1.2. Hyperboliccurvesandstablereduction. Toaddflexibilitytoourtreatmentofstablereduction,so
astoincludecurvesofsmallgenus,wewillallowpunctures D ⊂X(K)and D ⊂Y(K),whicharefinite
X Y
setsofType-I points suchthat f−1(D )= D . Recall thatasmooth,proper, puncturedcurve (X,D ) is
Y X X
said tobehyperbolicif χ(X,D )<0, where χ(X,D )=2−2g(X)−|D |is theEuler characteristic. A
X X X
semi-stableformalmodelofthepuncturedcurve(X,D )isanintegralproperadmissibleformalR-scheme
X
XsuchthatitsgenericfibreasanadicspaceisisomorphictoXad,andmoreover
• X isareducedconnectedcurveoverk withatmostordinarydoublepointsforsingularities,
s
• allpointsin D reducetodistinctsmoothpointsonX .
X s
The category Formss consists of semi-stable formal models of (X,D ), together with an isomorphism
X X
betweentheadicgenericfibreand Xad. Amorphismbetweentwosuchmodelsisamorphismofformal
R-schemes thatinduces theidentity on Xad. When(X,D ) is hyperbolic, thereexists aunique minimal
X
suchmodel,whichwecallthestableformalmodel. Itischaracterisedbythepropertiesofsemi-stability,
togetherwith theadditional constraint thattheEuler characteristic of every rational component ofthe
specialfibreX ,minusitssingularlocusandthesetofpunctures,isnegative.
s
1.3. Coleman’swideopenspaces. WerecallthenotionofwideopenballsandannuliduetoColeman
[Col89], in the language of adic spaces. A wide open ball is an adic space which is isomorphic to the
complement of the set |t| = 1 in the open ball Spa(K〈t〉,R〈t〉). A wide open annulus is an adic space
isomorphictothecomplementinawideopenballoftheopenset|t|≤pw forsome w∈R whichwe
>0
call thewidthof the annulus. We notethat a wide open ball possesses exactly one Type-Vpoint which
isnotthespecialisationofanyType-IIpoint. ThisType-Vpointiscalledtheapexpointofthewideopen
ball. Similarly, wideopenannulihaveexactly2suchpoints,whichwealsocallapexpoints.
1.4. The specialisation map. Let X be an admissible formal R-scheme with generic fibre Xad. As in
[Sch12,Theorem2.22]thespecialisationmapisamorphismoflocallyringedtopologicalspaces
sp : Xad,O+ → X,O ,
X (cid:16) Xad(cid:17) (cid:0) X(cid:1)
STABLEMODELSOFHECKEOPERATORS 5
whose fibres are called formal fibres. When X is semi-stable, it is possible to determine the nature of
the formal fibres by combining the work of Bosch-Lütkebohmert [BL85, Propositions 2.2 and 2.3] and
Berkovich[Ber90,Proposition2.4.4]. Weobtainthefollowingtheorem,seealso[BPR11,Theorem4.6].
Theorem1(Bosch–Lütkebohmert,Berkovich). LetξbeapointofX . Then
s
• ξisagenericpointifandonlyifsp−1(ξ)consistsofasingleType-IIpointofXad,
X
• ξisasmoothclosedpointifandonlyifsp−1(ξ)isawideopenball,
X
• ξisanordinarydoublepointifandonlyifsp−1(ξ)isawideopenannulus.
X
1.5. Semi-stablevertexsets. Asemi-stablevertexsetofthesmooth,proper,puncturedcurve(X,D )is
X
afinitesetV ofType-IIpointsofXad suchthat
• thespaceXad\V isadisjointunionofwideopenballsandfinitelymanywideopenannuli,
• thepointsin D belongtodistinctwideopenballsinXad\V.
X
ForaType-IIpoint x inasemi-stablevertexsetV,callitsvalencythenumberofapexpointsofwideopen
annuliinXad\V inthetopologicalclosureof x. ThecategoryVertss consistsofsemi-stablevertexsetsof
X
(X,D ),whereamorphismbetweentwosuchsetsisgivenbyinclusion. Theorem1allows ustoattach
X
toasemi-stableformalmodelXthefiniteset V :={sp−1(ξ)} ,whereξrangesoverthegenericpoints
X X ξ
of the irreducible components of X . It follows immediately from Theorem 1 that V is a semi-stable
s X
vertexsetfor(X,D ). ThisdefinesafunctorbetweenFormss andVertss. Itturnsoutthatitisinfactan
X X X
anti-equivalence ofcategories,whichisprovedin[BPR11,Theorem4.11].
Theorem2. ThefunctorFormss→Vertss : X7→V inducesananti-equivalenceofcategories.
X X X
Choosesemi-stableformalmodelsX,Yof(X,D )and(Y,D )respectively. Thisyieldsarationalmap
X Y
X¹¹ËYinducedby f. Wewill investigate whenthisrational map extendstoamorphismX→Y,and
whether we can arrange for such an extension to be a finite map. The equivalence we just established
gives us adirect way toinvestigatethisproblem, as semi-stable vertexsetsnaturally livein Xad,where
weunderstandthebehaviourof f. Thefollowingtheoremisprovedin[ABBR15,Theorem5.13].
Theorem3. LetX,Ybesemi-stable formalmodelsof(X,D )and(Y,D ),then f extendstoamorphism
X Y
X→Yifandonlyif f−1(V )⊆V . Thisextensionisfiniteifandonlyif f−1(V )=V .
Y X Y X
1.6. Faithfullyflatdescent. WewillbeinterestedinbasefieldsK thatarenotnecessarilyalgebraically
0
closedorcomplete. Wenowrecallsomerelevantpartsofdescenttheory,ofwhichaniceexpositionand
more details may be found in [ABBR15, Section 5]. Let (X,D) be a smooth, projective, geometrically
connected curve over a field K with a non-trivial non-Archimedean valuation of rank 1 and valuation
0
∧
ringR . Let K =K bethecompletionofanalgebraic closureof K ,whichisitselfalgebraically closed,
0 0 0
andletRbethevaluationringofK. Thetoolsoutlinedabovecanbeusedtoanalysesemi-stablemodels
of (X ,D ), after which we pass back to (X,D) by the theory of faithfully flat descent. More precisely,
K K
anysemi-stablemodelofX isisomorphictoX ,whereX isasemi-stablemodelofX oversomefinite
K R K
1
separableextensionK ofK . Moreover,thefollowingLemmaisprovedin[ABBR15,Lemma5.5].
1 0
Lemma1. Let f :X →Y beafinitemorphismbetweensmooth,projective,geometricallyconnectedcurves
over K , with semi-stable R -models X and Y respectively. Suppose that f extends to a finite morphism
0 0
X →Y ,then f alsoextendsuniquelytoafinitemorphismX →Y definedoverR .
R R 0
6 JANVONK
Proof. Therationalmap f :X ¹¹ËY extendstoamorphismifandonlyiftheprojectionπ:Γ →
f
X ofitsgraphΓ inX × Y isanisomorphism. Theisomorphismπ :Γ ≃Γ ×Spec(R)→X
f R R f f R
0 R
descendstoanisomorphismπ:Γ →Spec(R ),andhence f extendstoamorphismX →Y. This
f 0
extensionisnecessarilyfinite,asitbecomesfiniteafterfaithfullyflatbasechange. (cid:3)
2. STABLE MODELS OF CORRESPONDENCES
Inthissection,weproveananalogueforcorrespondencesofthestablereductiontheoremofDeligne–
Mumford [DM69]. This generalises the results for finite maps proved by Coleman [Col03] and Liu
[Liu06]. Ourproofusespropertiesofthespecialisationmapfromanalyticgeometryoutlinedabove.
2.1. Main definitions. Let K be a field equipped with a non-trivial non-Archimedean valuation |·| of
rank1,whosevaluationringRhasmaximalidealm. Apuncturedcorrespondenceisadiagram
π (X,D ) π
1 X 2
C:
(Y ,D ) (Y ,D )
1 1 2 2
where
• X,Y ,Y aresmooth,projective,geometricallyconnectedK-curves,
1 2
• π ,π arefiniteK-morphisms,
1 2
• D ,D ,D arefinitesetsofpunctureswithπ−1(D )=D =π−1(D ).
X 1 2 1 1 X 2 2
A punctured correspondence C is said to be hyperbolic if its objects are hyperbolic punctured curves in
thesenseofsection1.2. Asemi-stableR-modelofthepuncturedcorrespondenceCisadiagram
X
C:
Y Y
1 2
where
• X,Y andY areintegralflatpropersemi-stableR-curves,togetherwithisomorphismsX ≃
1 2 K
X as well as Y ≃Y and Y ≃Y , sothattheirformal completions along thespecial fibre
1,K 1 2,K 2
aresemi-stableformalR-modelsfor(X,D ),(Y ,D )and(Y ,D )inthesenseof1.2,
X 1 1 2 2
• themorphismsarefiniteandrestricttoπ ,π onthegenericfibres,viathegivenisomorphisms.
1 2
Wesaythatasemi-stableR-modelC dominatesanothersemi-stableR-modelC ,iftheobjectsofC dom-
1 2 1
inatetheobjectsofC pairwise. Asemi-stableR-modeliscalledstableifitisminimalwithrespecttothe
2
relation of domination. Thestable model of a hyperbolic correspondence is unique up to isomorphism
ifitexists,butingeneraltheobjectsofthestablemodelarenotthestablemodelsofitsobjects.
2.2. Stable models of Galois morphisms. Beforecoming toa proof of potentially stable reductionof
correspondences, which is thecontent ofTheorem A,weprove a lemma about Galois maps f :X →Y
to which thegeneral casewill bereduced. Theweaker statement, not insisting that theextensionof f
should be finite, was proved by Liu–Lorenzini [LL99, Proposition 4.4]. A proof for K = C is given in
p
Coleman[Col03,Section3],andtheproofweincludeiscloseinspirittothepaper[ABBR15].
Lemma2. Let f :(X,D )→(Y,D )beafiniteGaloismorphismbetweensmooth,projective,geometrically
X Y
connected, hyperbolic punctured curves over K. Assume (X,D ) has stable model X over R. Then there
X
existsauniquesemi-stablemodelY of(Y,D )suchthat f extendstoafinitemorphismX →Y.
Y
STABLEMODELSOFHECKEOPERATORS 7
Proof. Extend scalars to K∧. Let V ⊂ Xad be the stable vertex set of (X,D ), and set W = f(V).
X
Thereisaminimalsemi-stablevertexsetW′⊂Yadfor(Y,D )containingW,whichmaybeobtained
Y
byaddingafinitenumberofType-IIpointsifnecessary,see[ABBR15,Lemma3.15]. Wewillprove
thatW′ =W. Pick any element y ∈W′\W,and any element x ∈ f−1(y). We seethatthis means
that y mustbeofvalencyatleast3,whereas x mustbeofvalency2.
x y
f
Because the Galois action on fibres over Type-I points is transitive and Yad is connected, the map
f :Xad→Yad isaGalois coveringoftopologicalspaces. Itfollowsthatthevalencyof x mustbeat
leastthevalencyof y,whichisacontradiction. ThereforeW issemi-stable,andTheorem3assures
theexistenceofafinitemorphism f :X→Yofsemi-stableformalmodelsovertheringofintegral
∧
elementsinK ,whereXisinfactstable. WeobtainalgebraicmodelsX andY bythealgebraisation
theorem[Abb10,Corollaire2.3.19],andbothX andY descendtotheintegralclosureofRinsome
finiteseparable extensionof K,see[ABBR15,Lemma 5.4]. Finally, themorphism f descendstoa
finitemorphismoverthesamefieldbyfaithfullyflatdescentasinLemma1. (cid:3)
2.3. Remark. Wenotethatthistheoremisfalsewithouttheassumptionthat f isGalois. However,itis
truethatageneral f extendstoamorphismbetweenthestablemodelsofitssourceandtarget,although
suchanextensionwillingeneralfailtobefinite. See[LL99,Proposition4.4]formoredetails.
2.4. Potentiallystablereductionforcorrespondences. Wenowcometothemainresultofthispaper,
whichshouldbeviewed asananalogueofthetheoremofDeligne–Mumford [DM69,Corollary 2.7] on
potentially semi-stable reduction of smooth projective curves. We note that we already made essential
use of the result of Deligne–Mumford in the proof of Lemma 2, so we do not obtain a new proof by
specialisingtothedegeneratecasewhereourcorrespondenceisasinglecurvewithidentitymorphism.
TheoremA. LetC be a hyperbolicpunctured correspondence over K. Thereis a finiteseparable extension
ofK overwhichChasastableR-model,whichisuniqueuptoisomorphism.
∧
Proof. Change scalars to K . Consider theGalois closure g :(X,D)→(X,D )of bothπ and π ,
X 1 2
withrespecttosomeembeddingoftheirfunctionfieldsinto K(te).eThisyieldsadiagram
Galois (Y,D)
1 1
g π1
(X,D) (X,DX) π
e e 2
(Y,D)
Galois 2 2
where we set D= g−1(D ). We seethat (X,D) is hyperbolic, and as such there is a unique stable
X
vertex set V ⊂eXad by Deligne–Mumford [eDMe69, Corollary 2.7]. Set V := g(V) and W = π(V),
i i
which are esemi-estable vertex sets by Lemma 2. We must have V = π−1(W) bey transitivity of the
i i
Galois action on fibres, and hence π extends to a finite morphism between the corresponding
i
8 JANVONK
formal models. By the algebraisation theorem [Abb10, Corollaire 2.3.19], this yields three semi-
∧
stablecurvesovertheringofintegralelementsinK ,whichdescendtoafiniteseparableextension
of K by [ABBR15, Lemma 5.4]. We use faithfully flat descent as in Lemma 1 to show that the
morphismsdescendtothesameextension,givingusasemi-stablemodelC.
To show the existence of a stable model over the same field extension, we note that any semi-
stable model must have the property that the semi-stable vertex sets of its objects contain the
stable vertex sets. Hence, we may consider the stable vertex set S ⊂ Xad of (X,D ) and the semi-
X
stable vertex sets π(S)= T. Now repeat the following procedure: Enlarge S to contain π−1(T),
i i i i
and enlarge the sets T to contain π(S). This procedure terminates, as all the newly introduced
i i
Type-II points are necessarily contained in the finite semi-stable vertex sets corresponding to the
semi-stable model C constructed above via the Galois closure. Therefore this procedure yields a
semi-stablemodelC′ forCwhichisminimalwithrespecttotherelationofdomination. HenceC′ is
thestablemodelofC. (cid:3)
2.5. Remark. Wenotethatingeneral,thestablemodelofCdoesnotconsistofthestablemodelsofits
objects. AscanbeseenbelowforthecaseofHeckeoperators,typicallysomeextracomponentsappear.
Wealsonotethatcorrespondenceswhichfailtobehyperbolicstillhavepotentiallysemi-stablereduction,
butingeneraltherewillnotbeaminimalsuchmodel. Toovercomethis,wecanfixsemi-stablemodels
for all the objects of C, and run through the above procedure to construct their stable hull C, which is
theminimal semi-stable modelsuchthatalltheobjectsdominatetheircorresponding fixedsemi-stable
models. AsimilarrelativeresultisprovedinTheoremB.
3. WEIGHT-MONODROMY AND SKELETA OF CORRESPONDENCES
We now investigate how theknowledge of a semi-stable model fora correspondence C allows us to
deduceinformationaboutthespectrumofC. IfCisasemi-stablemodelofCthen,evenifitsmorphisms
failtobeflat,wemayusethenormalityofthesemi-stablemodelsinvolvedtoconstructfunctorialtrace
morphismsbylocalisationatheight1primesandreductiontotheflatcase. Letting f,g bethedefining
morphismsofY ,Y toSpec(R),thenweobtainafunctoriallinearisation
1 2
C∗:R1f Q →R1g Q ,
∗ l ∗ l
whichonK restrictstothemapC∗ onétalecohomology. InthecaseofHeckeoperators,whichwetreat
explicitlyinsection4,wehave Y =Y andtheeigenvalues areFouriercoefficientsofweighttwomod-
1 2
ular forms. Werecall theweight-monodromy filtrationonthecohomologygroupsofthegeneric fibres,
andtheconnectionwithsemi-stablemodels. Inoursetting,wededuceinformationonC∗ byinvestigat-
ing the properties of the geometric special fibre of C∗, which can be explicitly understood in terms of
combinatorialdatacomingfromthedualgraphsofthespecialfibres,andthecomponentsappearingin
thereductions. Therelevantnotionisthatofaskeletalsemi-stablemodelofacorrespondence.
Non-Noetherianmodels. AstheexampleoftheHeckeoperatorT onquaternionicShimuracurvesat
p
pin4.3shows,weknowthatinspiteoftheexistenceofastablemodelafterfiniteseparablebasechange
asinTheoremA,weareoftenforcedbothtopasstoaninfinitefieldextensionandallowinfinitelymany
components when we try to construct a skeletal such model. It might be natural to work in a larger
category instead, and we hope to generalise this material in the future to more general types of adic
spaces,soastoalsoincludesuchperfectoidspacesasLubin–Tatetowers[Wei12].
STABLEMODELSOFHECKEOPERATORS 9
3.1. The weight-monodromy filtration. We now recall the definition of the monodromy filtration
on the l-adic étale cohomology of a proper, smooth, separated scheme X of finite type over a non-
Archimedean local field K, and its relation with quantities that can be computed from a semi-stable
modelinthecaseofcurves. BythemonodromytheoremofGrothendieck[Gro72],thereexistsanilpo-
tentoperator
N ∈End(Hi (X ,Q ))
ét K l
such that every σ in a sufficiently small open subgroup of the inertia group I ⊂ Gal(Ks/K) acts as
exp(t (σ)N), where t : I → Z(1) is the maximal pro-l quotient map. We obtain an ascending mon-
l l l
odromyfiltrationM onHi (X ,Q ),characterisedbyNM ⊆M (−1)and
• ét K l i i−2
Ni :GrM −∼→GrM(−i)
i −i
FromthedescriptionoftheactionofinertiaintermsoftheoperatorN,weobtainawell-definedaction
ofthegeometricFrobeniusFr determinedbyunipotentinertia,andinparticularanotionofweights.
q
Conjecture 1 (Weight-monodromy conjecture). The eigenvalues of Fr on GrMHi (X ,Q ) are algebraic
q j ét K l
numberswhoseconjugatesallhavecomplexabsolutevaluesequaltoq(i+j)/2.
Let X be a semi-stable model for X with components Y in the special fibre, then the analysis of
i
thevanishing andnearbycyclesfunctorbyRapoport–Zink[RZ82]yields theconstructionoftheweight
spectralsequence,withfirstpage
Ep,q= Hq−2i Y[p+2i+1],Q (−i) ⇒ Hp+q X ,Q ,
1 M ét (cid:16) s l (cid:17) ét (cid:0) K l(cid:1)
i≥max(0,−p)
where Y[m] is thedisjoint union of the m-fold intersections of the Y. This induces a filtration on coho-
i
mologywhichisknowntocoincidewiththeweight-monodromyfiltrationforcurves,whereeverything
was proved by Grothendieck [Gro72]. Deligne proved the statement over function fields [Del71], and
somesubsequentresultsreducecasesinmixedcharacteristictothistheorem,seeScholze[Sch12].
3.2. Spectralpropertiesofcorrespondences. LetX beasmoothpropercurveoverK,thentheknowl-
edgeofanexplicitsemi-stablemodelX forX helpsuscomputethegradedpiecesoftheweightfiltration.
Indeed,theweightspectralsequence[RZ82]givesustheshortexactsequence
0→H1(X ,Q )→H1(X ,Q )→H1(Γ ,Q )(−1)→0,
ét s l ét K l X l
f
where Γ is the dual graph attached to X, and X denotes the normalisation of its geometric special
X s
fibre. Let J be the Néron model of the Jacobianfof X. The quotient of J by its identity connected
s
componentJ0 definestheNéroncomponentgroupΦ,sowegettheexactsequence
s
0→J0→J →Φ→0.
s s
Recall that the results of [Ray70] imply that J0 ≃ Pic0(X ), see [Gro72, Exposé IX, 12.1.11]. This
s s
meanswecanfurtherdecomposeJ0 asfollows
s
0→H1(Γ ,Z)⊗G →J0→Pic0(X )→0.
X m s s
f
NotethatweneednotassumethatX isregular,see[Rib90,Section2]. ThealgebraicgroupH1(Γ ,Z)⊗
X
G iscalledthetoricpartofJ. TheNéroncomponentgroupΦisisomorphictothecokernelofthemap
m
fromthetoricparttoitsdual,givenbythemonodromypairing[Gro72,Théorème11.5].
10 JANVONK
Weseethatanexplicitknowledgeofasemi-stablemodelCforCisexactlywhatisneededtodescribe
theinduced map on thefirst step of the weight-monodromy filtration. Thereduction of Cdecomposes
as a finite set of correspondences between irreducible curves over the residue field of R, providing a
geometric description of the induced morphism Pic0(Y ) → Pic0(Y ). For the case of the Hecke
1,s 2,s
operator T on modular curves XB treated in 4.3, thiseyields the celeebrated Eichler–Shimura relation
p 0
T =Frob+Ver. Todescribetheinducedmorphismbetweenboththegradedpiecesofweight2andthe
p
Néron component groups Φ →Φ , we use Raynaud’s theorem and the monodromy pairing to reduce
1 2
theproblem tofinding a geometricinterpretation of themaps betweenthetoricparts of theJacobians.
As recalled above, we may describe the toric part in terms of the dual graph of a semi-stable model,
whichhasanaturalinterpretationasaskeletoninthesettingofadicspaces. Wenowrecalltherelevant
theory,anduseittomotivatethedefinitionofaskeletalsemi-stablemodelin3.7.
3.3. The skeletonof a semi-stable vertexset. Toasemi-stable vertex set V for(X,D ),wecanasso-
X
ciate a combinatorial structure which we call the skeleton Σ of X with respect to V. First, we define
V
Γ as thesetof points of Xad that are notcontained ina wideopenball which is disjoint from V ∪D ,
V X
endowedwith thesubspacetopology. Recall thatthemaximal Hausdorffquotient Γ ,oftenreferredto
V
as the Berkovich skeleton of X with respect to V, can naturally be viewed as a metric realisation of the
dual graph. Itsvertex setis V ∪D ,andedgescomeintwoflavours: Thereisanedge e betweentwo
X Q
vertices in V forevery intersectionpointQ ofthecorresponding components in thespecial fibre ofX ,
V
andwesetthelength l(e )ofe tobethewidthofthewideopenannulus sp−1(Q). Everyvertexin D
Q Q V X
is adjacent toexactly oneedgeof length ∞. Define theskeleton of Xad with respectto V,or X ,to be
V
thepairΣ =(Γ ,L ),where L ={l(e):e edgeofΓ }consistsofthesetofedgelengthsappearing
V V V V V
inthemaximalHausdorffquotientΓ ,viewedasametricgraph. WedefinethegenusofaType-IIpoint
V
x inΓ tobethegenusoftheresiduefieldk(x)asdefinedin1.1.
V
3.4. Example. Considertheprojectiveclosureof y2=x3+x2+p3,overO ,andletX betheblow-up
C
p
atasmoothpointofthespecialfibre. ThenX isanormalsemi-stablemodelwhosespecialfibreconsists
ofanodalcurve C andaprojectiveline C ,crossingtransversally. Nowset X tobethegenericfibreof
1 2
X,andD={(0,1,0)}thepointatinfinity. TheformalcompletionXofX isasemi-stableformalmodel
for(X,D),anditscorrespondingskeletonΣ canbevisualisedas
X
C
2
Type-V:
1 D
sp
X
Type-V: ∞
C
1
3
3.5. Harmonic morphisms. Given a finite morphism f : X → Y of semi-stable formal models for
(X,D ) and (Y,D ), we know that f−1(V ) = V . In general, it is not true that f−1(Γ ) = Γ , see
X Y Y X Y X
[ABBR15,Remark 5.23]. Whenthis holds, wesay that f is a skeletalfinitemorphism. Formaps f,we
canalwaysmodify f soastomakeitskeletal,see[CKK15,Theorem2.4]and[ABBR15,Corollary4.18].
Let x be an apex point of a wide open annulus in Xad\V , and set l to be the width of the annulus.
X x
It can be shown that the ratio d := l /l equals the ramification index of the map f : X → Y at
x f(x) x s s
the point sp (x), see [Col05, Lemma 2.1]. We use this to define d for all apex points attached to X,
X x